Elsevier

Automatica

Volume 46, Issue 1, January 2010, Pages 167-173
Automatica

Brief paper
Noncausal linear periodically time-varying scaling for robust stability analysis of discrete-time systems: Frequency-dependent scaling induced by static separators

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Abstract

This article is concerned with robust stability analysis of discrete-time systems and introduces a novel and powerful technique that we call noncausal linear periodically time-varying (LPTV) scaling. Based on the discrete-time lifting together with the conventional but general scaling approach, we are led to the notion of noncausal LPTV scaling for LPTV systems, and its effectiveness is demonstrated with a numerical example. To separate the effect of noncausal and LPTV characteristics of noncausal LPTV scaling to see which is a more important source leading to the effectiveness, we then consider the case of LTI systems as a special case. Then, we show that even static noncausal LPTV scaling has an ability of inducing frequency-dependent scaling when viewed in the context of the conventional LTI scaling, while causal LPTV scaling fails to do so. It is further discussed that the effectiveness of noncausal characteristics leading to the frequency-domain interpretation can be exploited even for LPTV systems by considering the νN-lifted transfer matrices of N-periodic systems.

Introduction

It is well known that the small-gain theorem plays a fundamental role in robust stability analysis and μ-analysis (Packard & Doyle, 1993) is effective when we deal with structured uncertainties. The latter employs structured frequency-dependent scaling according to the structure of uncertainties, but it is cumbersome to get suitable frequency-dependent scaling (see, e.g., Zhou and Doyle (1998)). On the other hand, this article aims at investigating the use of the discrete-time lifting technique in robust stability analysis, and introduces a natural notion that accompanies such treatment, which we call noncausal linear periodically time-varying (LPTV) scaling.

In contrast to the conventional frequency-domain scaling, this new scaling can be regarded as a sort of time-domain scaling, and can be effective even for unstructured uncertainties. More importantly, we show that when the system is LTI, it can be regarded as inducing frequency-dependent scaling if it is interpreted in the conventional context in the original time axis without lifting.To put it another way, by using noncausal LPTV scaling, we can, in a sense, convert the search of frequency-dependent scaling into that of static scaling by the discrete-time lifting technique. In such a connection, it should be noted that noncausal LPTV scaling generalizes the conventional LTI scaling in two ways, i.e., (i) by introducing LPTV characteristics and (ii) by allowing noncausal operations in scaling. Regarding these two aspects, we show that (ii) is the most important source for the effectiveness of noncausal LPTV scaling in the frequency domain.

The contents of this article are as follows. We first describe the lifting-based robust stability analysis of discrete-time systems in Section 2. Based on the lifting-based treatment, we present in Section 3 the basic ideas for noncausal LPTV scaling. A numerical example with an LPTV nominal system is given there, which shows that noncausal LPTV scaling is indeed effective for robust stability analysis of discrete-time systems, and also motivates the subsequent discussions. In Section 4, we confine ourselves to the case with LTI systems, and give some theoretical results that suggest the effectiveness of noncausal LPTV scaling. In particular, we show that even static noncausal LPTV scaling has an ability of inducing frequency-dependent scaling if it is interpreted in the conventional lifting-free treatment. It is also shown that this promising property can be exploited even when the systems are LPTV rather than LTI, if we consider the νN-lifted transfer matrices of N-periodic systems. Section 5 summarizes the arguments of the article and gives some remarks on further research directions.

Notation: N denotes the set of positive integers.

Section snippets

Robust stability of discrete-time closed-loop systems

Consider the discrete-time system shown in Fig. 1 with the nominal system G and the uncertainty Δ. We assume that G is a q1-input q2-output, internally stable, finite-dimensional (FD) LPTV system with period N (i.e., an N-periodic system), where NN. We assume that ΔΔ for some given set Δ (possibly consisting of static systems only) satisfying the assumption:

A1 Every ΔΔ is FD, N-periodic, internally stable, and Δ is a connected set such that 0Δ.

We sometimes consider the case when every ΔΔ

Noncausal LPTV scaling and a motivating numerical example

In this section, we give a brief idea on the technique that we introduce in this article, i.e., noncausal LPTV scaling. We consider a simple and special case of noncausal LPTV scaling, apply it to a numerical example, and demonstrate its effectiveness. The aim of this section, however, rather lies in motivating the theoretical discussions in this article.

Assuming q1=q2 here, let us consider a typical separator of the form Θ̂(z)=diag[γ2Ŵ(z)Ŵ(z),Ŵ(z)Ŵ(z)],γ>0 corresponding to the D

Properties of noncausal LPTV scaling applied to LTI systems

For the reason stated in the preceding section, we confine ourselves to the case when G and Δ are LTI in this section. Then G and Δ can be associated with their transfer matrices G(ζ) and Δ(ζ), respectively, as well as their N-lifted transfer matrices Ĝ(z) and Δ̂(z) for NN taken arbitrarily. Note that the forward shift in time k is denoted by ζ to distinguish it from the symbol z for that in the lifted domain (i.e., z corresponds to ζN). Obviously, Proposition 1 holds under the replacement of

Conclusion

This article studied the use of lifting in the robust stability analysis of discrete-time systems, which naturally leads to a novel technique called noncausal LPTV scaling. A numerical example with an LPTV nominal system was first studied to show the effectiveness of such a technique. We then discussed the properties of noncausal LPTV scaling applied to LTI systems, and showed that it has an important interpretation. That is, we showed that even static noncausal LPTV scaling induces

Tomomichi Hagiwara was born in Osaka, Japan on March 28, 1962. He received his B.E., M.E. and Ph.D. degrees in electrical engineering from Kyoto University, Kyoto, Japan in 1984, 1986 and 1990, respectively. Since 1986, he has been with the Department of Electrical Engineering, Kyoto University, where he is Professor since 2001. His research interests include dynamical system theory and control theory such as analysis and design of sampled-data systems, time-delay systems and

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    This generally results in conservativeness in robust stability analysis. For reducing the conservativeness, discrete-time noncausal linear periodically time-varying (LPTV) scaling was introduced in Hagiwara and Ohara (2010) recently. This approach can be naturally introduced by employing the separator-type robust stability theorem under the lifting treatment (Bittanti & Colaneri, 2009).

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Tomomichi Hagiwara was born in Osaka, Japan on March 28, 1962. He received his B.E., M.E. and Ph.D. degrees in electrical engineering from Kyoto University, Kyoto, Japan in 1984, 1986 and 1990, respectively. Since 1986, he has been with the Department of Electrical Engineering, Kyoto University, where he is Professor since 2001. His research interests include dynamical system theory and control theory such as analysis and design of sampled-data systems, time-delay systems and two-degree-of-freedom control systems.

Yasuhiro Ohara was born in Hyogo, Japan on January 28, 1985. He received his B.E. and M.E degrees in electrical engineering from Kyoto University, Kyoto, Japan in 2007 and 2009, respectively. Since 2009, he has been with Honda R&D Company, Tochigi, Japan.

The material in this article was partially presented at the IFAC Workshop on PSYCO ’07, August 29–31, 2007, St. Petersburg and the IFAC World Congress 2008, July 6–11, 2008, Seoul. This article was recommended for publication in revised form by Associate Editor Lihua Xie under the direction of Editor Roberto Tempo.

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